Timeline for Continuous self-maps in the Golomb space that are neither increasing nor decreasing
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2017 at 0:17 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Added a link to arXiv paper in the Remark
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| Nov 25, 2017 at 8:46 | comment | added | Taras Banakh | @YCor This is a bit long. I will add the proof of this fact to the update of the corresponding paper in arXiv. | |
| Nov 24, 2017 at 18:19 | comment | added | YCor | Can you elaborate about your new remark (that there are $2^{\aleph_0}$ self-maps)? | |
| Nov 24, 2017 at 17:08 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Removed mentioning of rigidity of the Golomb space
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| Nov 22, 2017 at 21:54 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Turned problem into Remark (as it has been solved)
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| Nov 22, 2017 at 12:26 | history | edited | Taras Banakh | CC BY-SA 3.0 |
added 1 character in body
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| Nov 21, 2017 at 14:30 | vote | accept | Dominic van der Zypen | ||
| Nov 21, 2017 at 8:23 | history | edited | Taras Banakh | CC BY-SA 3.0 |
edited the definition of the set $Y$
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| Nov 21, 2017 at 8:22 | comment | added | Taras Banakh | @YCor Thank for your comment. Indeed, the set $Y$ had wrong definition. Now it is fixed. Concerning the cardinality of the set of self-maps of the Golomb space I am sure that it is continuum (the number of branches of some recursively defined tree of partial finite functions) but the proof of this fact still escapes from me (I hope not for long time). | |
| Nov 21, 2017 at 8:18 | history | edited | Taras Banakh | CC BY-SA 3.0 |
edited body
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| Nov 21, 2017 at 8:12 | comment | added | YCor | A smaller basis for the topology consists of the $a+b\mathbb{N}_0$ for $a,b$ coprime and $1\le a<b$ (i.e. I discard those with "holes" such as $3+2\mathbb{N}_0$). Using this basis makes life simpler and avoids an unnecessary argument involving finite fibers. (btw you probably mean $Y=(f(x)+b\mathbb{Z})\cap (\mathbb{N}\smallsetminus (f(x)+b\mathbb{N}_0))$) | |
| Nov 21, 2017 at 8:00 | comment | added | YCor | This argument works for every $f:\mathbb{N}\to\mathbb{N}$ such that $n|f(n)$ for all $n$ and $b|f(n+b)-f(n)$ for all $n,b$. How many such maps are there? | |
| Nov 21, 2017 at 7:57 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Added a problem
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| Nov 21, 2017 at 7:50 | history | edited | YCor | CC BY-SA 3.0 |
added "no constant term" to clarify
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| Nov 21, 2017 at 7:39 | history | edited | Taras Banakh | CC BY-SA 3.0 |
deleted 148 characters in body
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| Nov 21, 2017 at 6:50 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Rewrote the answer in the form Theorem-Corollary.
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| Nov 21, 2017 at 6:44 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Rewrote the answer in the form Theorem-Corollary.
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| Nov 21, 2017 at 6:34 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Rewrote the answer in the form Theorem-Corollary.
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| Nov 21, 2017 at 1:53 | history | answered | Taras Banakh | CC BY-SA 3.0 |